# Properties

 Label 2-43-43.36-c5-0-7 Degree $2$ Conductor $43$ Sign $0.698 - 0.715i$ Analytic cond. $6.89650$ Root an. cond. $2.62611$ Motivic weight $5$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + 7.09·2-s + (1.12 + 1.94i)3-s + 18.4·4-s + (51.2 + 88.8i)5-s + (7.96 + 13.8i)6-s + (−36.3 + 62.9i)7-s − 96.5·8-s + (118. − 206. i)9-s + (364. + 630. i)10-s + 469.·11-s + (20.6 + 35.7i)12-s + (235. − 408. i)13-s + (−257. + 446. i)14-s + (−115. + 199. i)15-s − 1.27e3·16-s + (−406. + 704. i)17-s + ⋯
 L(s)  = 1 + 1.25·2-s + (0.0720 + 0.124i)3-s + 0.575·4-s + (0.917 + 1.58i)5-s + (0.0903 + 0.156i)6-s + (−0.280 + 0.485i)7-s − 0.533·8-s + (0.489 − 0.848i)9-s + (1.15 + 1.99i)10-s + 1.16·11-s + (0.0414 + 0.0717i)12-s + (0.386 − 0.669i)13-s + (−0.351 + 0.609i)14-s + (−0.132 + 0.228i)15-s − 1.24·16-s + (−0.341 + 0.591i)17-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.698 - 0.715i)\, \overline{\Lambda}(6-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.698 - 0.715i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$43$$ Sign: $0.698 - 0.715i$ Analytic conductor: $$6.89650$$ Root analytic conductor: $$2.62611$$ Motivic weight: $$5$$ Rational: no Arithmetic: yes Character: $\chi_{43} (36, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 43,\ (\ :5/2),\ 0.698 - 0.715i)$$

## Particular Values

 $$L(3)$$ $$\approx$$ $$2.99186 + 1.26112i$$ $$L(\frac12)$$ $$\approx$$ $$2.99186 + 1.26112i$$ $$L(\frac{7}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad43 $$1 + (1.21e4 - 673. i)T$$
good2 $$1 - 7.09T + 32T^{2}$$
3 $$1 + (-1.12 - 1.94i)T + (-121.5 + 210. i)T^{2}$$
5 $$1 + (-51.2 - 88.8i)T + (-1.56e3 + 2.70e3i)T^{2}$$
7 $$1 + (36.3 - 62.9i)T + (-8.40e3 - 1.45e4i)T^{2}$$
11 $$1 - 469.T + 1.61e5T^{2}$$
13 $$1 + (-235. + 408. i)T + (-1.85e5 - 3.21e5i)T^{2}$$
17 $$1 + (406. - 704. i)T + (-7.09e5 - 1.22e6i)T^{2}$$
19 $$1 + (548. + 950. i)T + (-1.23e6 + 2.14e6i)T^{2}$$
23 $$1 + (1.42e3 + 2.47e3i)T + (-3.21e6 + 5.57e6i)T^{2}$$
29 $$1 + (-3.90e3 + 6.76e3i)T + (-1.02e7 - 1.77e7i)T^{2}$$
31 $$1 + (-1.20e3 - 2.09e3i)T + (-1.43e7 + 2.47e7i)T^{2}$$
37 $$1 + (3.72e3 + 6.45e3i)T + (-3.46e7 + 6.00e7i)T^{2}$$
41 $$1 - 1.52e4T + 1.15e8T^{2}$$
47 $$1 + 1.53e4T + 2.29e8T^{2}$$
53 $$1 + (-4.74e3 - 8.22e3i)T + (-2.09e8 + 3.62e8i)T^{2}$$
59 $$1 - 3.48e4T + 7.14e8T^{2}$$
61 $$1 + (2.14e4 - 3.72e4i)T + (-4.22e8 - 7.31e8i)T^{2}$$
67 $$1 + (1.94e4 + 3.37e4i)T + (-6.75e8 + 1.16e9i)T^{2}$$
71 $$1 + (3.44e3 - 5.96e3i)T + (-9.02e8 - 1.56e9i)T^{2}$$
73 $$1 + (-1.01e4 + 1.76e4i)T + (-1.03e9 - 1.79e9i)T^{2}$$
79 $$1 + (1.09e4 - 1.89e4i)T + (-1.53e9 - 2.66e9i)T^{2}$$
83 $$1 + (-6.52e3 - 1.13e4i)T + (-1.96e9 + 3.41e9i)T^{2}$$
89 $$1 + (1.67e4 + 2.90e4i)T + (-2.79e9 + 4.83e9i)T^{2}$$
97 $$1 - 6.64e4T + 8.58e9T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$